MATLAB實例：多元函數擬合(線性與非線性)

做者：凱魯嘎吉 - 博客園 http://www.cnblogs.com/kailugaji/html

如今用最小二乘法擬合多元函數，實現線性擬合與非線性擬合，其中非線性擬合要求自定義擬合函數。3d

下面給出三種擬合方式，第一種是多元線性擬合(迴歸)，第二三種是多元非線性擬合，實際中第二三種方法是一個意思，任選一種便可，推薦第二種擬合方法。code

1. MATLAB程序

fit_nonlinear_data.m

function [beta, r]=fit_nonlinear_data(X, Y, choose) % Input: X 自變量數據(N, D)， Y 因變量(N, 1)，choose 1-regress, 2-nlinfit 3-lsqcurvefit if choose==1 X1=[ones(length(X(:, 1)), 1), X]; [beta, bint, r, rint, states]=regress(Y, X1) % 多元線性迴歸 % y=beta(1)+beta(2)*x1+beta(3)*x2+beta(4)*x3+... % beta—係數估計 % bint—係數估計的上下置信界 % r—殘差 % rint—診斷異常值的區間 % states—模型統計信息 rcoplot(r, rint) saveas(gcf,sprintf('線性曲線擬合_殘差圖.jpg'),'bmp'); elseif choose==2 beta0=ones(7, 1); % 初始值的選取可能會致使結果具備較大的偏差。 [beta, r, J]=nlinfit(X, Y, @myfun, beta0) % 非線性迴歸 % beta—係數估計 % r—殘差 % J—雅可比矩陣 [Ypred,delta]=nlpredci(@myfun, X, beta, r, 'Jacobian', J) % 非線性迴歸預測置信區間 % Ypred—預測響應 % delta—置信區間半角 plot(X(:, 1), Y, 'k.', X(:, 1), Ypred, 'r'); saveas(gcf,sprintf('非線性曲線擬合_1.jpg'),'bmp'); elseif choose==3 beta0=ones(7, 1); % 初始值的選取可能會致使結果具備較大的偏差。 [beta,resnorm,r, ~, ~, ~, J]=lsqcurvefit(@myfun,beta0,X,Y) % 在最小二乘意義上解決非線性曲線擬合（數據擬合）問題 % beta—係數估計 % resnorm—殘差的平方範數 sum((fun(x,xdata)-ydata).^2) % r—殘差 r=fun(x,xdata)-ydata % J—雅可比矩陣 [Ypred,delta]=nlpredci(@myfun, X, beta, r, 'Jacobian', J) plot(X(:, 1), Y, 'k.', X(:, 1), Ypred, 'r'); saveas(gcf,sprintf('非線性曲線擬合_2.jpg'),'bmp'); end end function yy=myfun(beta,x) %自定義擬合函數 yy=beta(1)+beta(2)*x(:, 1)+beta(3)*x(:, 2)+beta(4)*x(:, 3)+beta(5)*(x(:, 1).^2)+beta(6)*(x(:, 2).^2)+beta(7)*(x(:, 3).^2); end

demo.m

clear clc X=[1 13 1.5; 1.4 19 3; 1.8 25 1; 2.2 10 2.5;2.6 16 0.5; 3 22 2; 3.4 28 3.5; 3.5 30 3.7]; Y=[0.330; 0.336; 0.294; 0.476; 0.209; 0.451; 0.482; 0.5]; choose=1; fit_nonlinear_data(X, Y, choose)

2. 結果

(1)多元線性擬合(regress)

choose=1:orm

>> demo beta = 0.200908829282537 0.044949392540298 -0.003878606875016 0.070813489681112 bint = -0.026479907290565 0.428297565855639 -0.057656451966002 0.147555237046598 -0.017251051845827 0.009493838095795 0.000201918738160 0.141425060624065 r = 0.028343433030705 -0.066584917256987 0.038333946339215 0.037954851676187 -0.082126284727611 0.058945364984698 -0.010982985302994 -0.003883408743214 rint = -0.151352966773048 0.208039832834458 -0.188622801533810 0.055452967019837 -0.090283529625345 0.166951422303776 -0.090266067743345 0.166175771095720 -0.108068661106325 -0.056183908348897 -0.130409602930181 0.248300332899576 -0.206254481234707 0.184288510628719 -0.184329400080620 0.176562582594191 states = 0.768591079367914 4.428472778943478 0.092289917768436 0.004625488283939

(2)多元非線性擬合(nlinfit)

choose=2:htm

>> demo beta = 0.312525876099987 0.015300533415459 -0.036942272680920 0.299760796634952 0.009412595106141 0.000976411370591 -0.062931846673372 r = 1.0e-03 * -0.047521336834000 0.127597019984715 -0.092883949615763 -0.040370056416994 0.031209476614974 0.211856736183458 -0.727835090583939 0.537947200592082 J = 1.0e+02 * 0.010000000000266 0.010000000001236 0.129999999998477 0.014999999999641 0.010000000007909 1.689999999969476 0.022499999999756 0.010000000000266 0.014000000006524 0.189999999999301 0.029999999999283 0.019600000006932 3.609999999769248 0.089999999999024 0.010000000000266 0.018000000011811 0.249999999990199 0.009999999999965 0.032399999999135 6.250000000033778 0.010000000000377 0.009999999999679 0.022000000005116 0.099999999998065 0.025000000000218 0.048400000003999 1.000000000103046 0.062500000001264 0.009999999999972 0.025999999998421 0.159999999998889 0.004999999999982 0.067599999997174 2.559999999999039 0.002499999999730 0.009999999999679 0.029999999991726 0.219999999999713 0.019999999999930 0.089999999993269 4.839999999890361 0.040000000000052 0.009999999999092 0.033999999985031 0.279999999990611 0.034999999998348 0.115599999997155 7.839999999636182 0.122500000000614 0.010000000000266 0.034999999992344 0.299999999994194 0.037000000000420 0.122499999994626 8.999999999988553 0.136899999999292 Ypred = 0.330047521336834 0.335872402980015 0.294092883949616 0.476040370056417 0.208968790523385 0.450788143263817 0.482727835090584 0.499462052799408 delta = 0.011997285626178 0.011902559677366 0.011954353934643 0.012001513980794 0.012005923574387 0.011706970437467 0.007666390995581 0.009878186927507

(3)多元非線性擬合(lsqcurvefit)

choose=3:blog

>> demo beta = 0.312525876070457 0.015300533464733 -0.036942272680581 0.299760796608728 0.009412595094407 0.000976411370579 -0.062931846666179 resnorm = 8.937848643213721e-07 r = 1.0e-03 * 0.047521324135769 -0.127597015215197 0.092883952947764 0.040370060121864 -0.031209466218374 -0.211856745335304 0.727835089662676 -0.537947200236699 J = 1.0e+02 * (1,1) 0.010000000000000 (2,1) 0.010000000000000 (3,1) 0.010000000000000 (4,1) 0.010000000000000 (5,1) 0.010000000000000 (6,1) 0.010000000000000 (7,1) 0.010000000000000 (8,1) 0.010000000000000 (1,2) 0.010000000000000 (2,2) 0.014000000059605 (3,2) 0.017999999970198 (4,2) 0.022000000029802 (5,2) 0.026000000014901 (6,2) 0.030000000000000 (7,2) 0.034000000059605 (8,2) 0.035000000000000 (1,3) 0.130000000000000 (2,3) 0.190000000000000 (3,3) 0.250000000000000 (4,3) 0.100000000000000 (5,3) 0.160000000000000 (6,3) 0.220000000000000 (7,3) 0.280000000000000 (8,3) 0.300000000000000 (1,4) 0.015000000000000 (2,4) 0.030000000000000 (3,4) 0.010000000000000 (4,4) 0.025000000000000 (5,4) 0.005000000000000 (6,4) 0.020000000000000 (7,4) 0.035000000000000 (8,4) 0.036999999880791 (1,5) 0.010000000000000 (2,5) 0.019599999934435 (3,5) 0.032399999983609 (4,5) 0.048400000035763 (5,5) 0.067599999997765 (6,5) 0.090000000000000 (7,5) 0.115600000023842 (8,5) 0.122500000000000 (1,6) 1.690000000000000 (2,6) 3.610000000000000 (3,6) 6.250000000000000 (4,6) 1.000000000000000 (5,6) 2.560000000000000 (6,6) 4.840000000000000 (7,6) 7.840000000000000 (8,6) 9.000000000000000 (1,7) 0.022500000000000 (2,7) 0.090000000000000 (3,7) 0.010000000000000 (4,7) 0.062500000000000 (5,7) 0.002500000000000 (6,7) 0.040000000000000 (7,7) 0.122500000000000 (8,7) 0.136899999976158 Ypred = 0.330047521324136 0.335872402984785 0.294092883952948 0.476040370060122 0.208968790533782 0.450788143254665 0.482727835089663 0.499462052799763 delta = 0.011997285618724 0.011902559623756 0.011954353977139 0.012001513949620 0.012005923574975 0.011706970418735 0.007666391016173 0.009878186931566

注意：多元非線性函數擬閤中參數的初始值須要提早設置，有些狀況下，參數的初始選取對函數擬合結果影響極大，須要謹慎處理。第二三種方法中，因爲數據是多維的，所以只展現了第一個維度的擬合函數圖。若有須要，可自行修改。ci